A proper orthogonal decomposition approach to microstructure model reduction in Rodrigues space with applications to optimal control of microstructure-sensitive properties

A novel reduced-order modeling approach to represent crystallographic texture in polycrystals is presented. This method involves generation of a reduced orthogonal basis for representing polycrystal texture in Rodrigues space. The reduced basis is generated from the textures obtained from different deformation modes (shear, uniaxial tension, etc.) using proper orthogonal decomposition and the method of snapshots. This model reduction can account for all essential information needed to describe the microstructure and leads to tremendous savings in computational time. Based on the above method a multiscale model for microstructure evolution during deformation is proposed driven by the macroscale velocity gradient. A continuum sensitivity-based gradient optimization framework is introduced in the above model for control of microstructure-sensitive material properties. Emphasis is given to relevant design problems that demonstrate the effectiveness of the above method for optimal control of texture-dependent material properties during deformation such as the magnetic hysteresis losses, the yield stress and the Lankford parameter R.